Final answer:
Electrons are paired up in hexaaquacobalt(III) because the ligand field created by the water molecules is strong enough to cause a significant octahedral crystal field splitting, leading to a low-spin complex where electron pairing is energetically favored.
Step-by-step explanation:
The reason why electrons are paired up in hexaaquacobalt(III) is due to the nature of the ligands and the resulting crystal field splitting. In this case, water molecules are acting as ligands, which can be either a weak or strong field ligand, depending on the metal ion they are bound to.
For Co³⁺, the ligand field is strong enough to cause a large separation between the energy levels of the d-orbitals (t2g and eg), which is denoted by a large Δoct (octahedral crystal field splitting parameter).
This large splitting means electron pairing is energetically favorable compared to promoting an electron to the higher energy eg orbital, leading to a low-spin complex where all electrons are paired. Thus, hexaaquacobalt(III), similar to low-spin [Fe(CN)6]3−, will have its electrons paired due to this significant crystal field splitting.